JHEP11(2017)137
Published for SISSA by SpringerReceived: August 31, 2017 Accepted: November 16, 2017 Published: November 22, 2017
Surface operators, chiral rings and localization in
N = 2 gauge theories
S.K. Ashok,a,b M. Bill`o,c,d E. Dell’Aquila,c M. Frau,c,d V. Gupta,a,b R.R. Johna,b and A. Lerdae,d
aInstitute of Mathematical Sciences, C.I.T. Campus,
Taramani, Chennai 600113, India
bHomi Bhabha National Institute, Training School Complex, Anushakti Nagar,
Mumbai 400085, India
cUniversit`a di Torino, Dipartimento di Fisica,
Via P. Giuria 1, 10125 Torino, Italy
dArnold-Regge Center and INFN — Sezione di Torino,
Via P. Giuria 1, 10125 Torino, Italy
eUniversit`a del Piemonte Orientale, Dipartimento di Scienze e Innovazione Tecnologica
Viale T. Michel 11, 15121 Alessandria, Italy
E-mail: sashok@imsc.res.in,billo@to.infn.it,edellaquila@gmail.com,
frau@to.infn.it,varungupta@imsc.res.in,renjan@imsc.res.in,
lerda@to.infn.it
Abstract: We study half-BPS surface operators in supersymmetric gauge theories in four and five dimensions following two different approaches. In the first approach we analyze the chiral ring equations for certain quiver theories in two and three dimensions, coupled respectively to four- and five-dimensional gauge theories. The chiral ring equations, which arise from extremizing a twisted chiral superpotential, are solved as power series in the infrared scales of the quiver theories. In the second approach we use equivariant localization and obtain the twisted chiral superpotential as a function of the Coulomb moduli of the four- and five-dimensional gauge theories, and find a perfect match with the results obtained from the chiral ring equations. In the five-dimensional case this match is achieved after solving a number of subtleties in the localization formulas which amounts to choosing a particular residue prescription in the integrals that yield the Nekrasov-like partition functions for ramified instantons. We also comment on the necessity of including Chern-Simons terms in order to match the superpotentials obtained from dual quiver descriptions of a given surface operator.
Keywords: Supersymmetry and Duality, Duality in Gauge Field Theories, Extended Supersymmetry
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Contents
1 Introduction 1
2 Twisted superpotential for coupled 2d/4d theories 3
2.1 SU(2) 4
2.2 Twisted chiral ring in quiver gauge theories 7
2.3 SU(3) 12
3 Twisted superpotential for coupled 3d/5d theories 14
3.1 Twisted chiral ring in quiver gauge theories 14
3.2 SU(2) and SU(3) 18
4 Ramified instantons in 4d and 5d 20
4.1 Localization in 4d 21
4.2 Localization in 5d 26
5 Superpotentials for dual quivers 27
5.1 Adding Chern-Simons terms 31
6 Conclusions and perspectives 32
A Chiral correlators in 5d 33
B Chern-Simons terms in an SU(4) example 34
1 Introduction
In this paper we study the low-energy effective action that governs the dynamics of half-BPS surface operators in theories with eight supercharges. We focus on pure SU(N ) theories in four dimensions and in five dimensions compactified on a circle, and explore their Coulomb branch where the adjoint scalars acquire a vacuum expectation value (vev).
In four dimensions, a surface defect supports on its world-volume a two-dimensional gauge theory that is coupled to the “bulk” four-dimensional theory, see [1] for a review. This combined 2d/4d system is described by two holomorphic functions: the prepotential F and the twisted superpotential W . The prepotential governs the dynamics of the bulk theory and depends on the Coulomb vev’s and the infra-red (IR) scale of the gauge theory in four dimensions. The twisted superpotential controls the two-dimensional dynamics on the surface operator, and is a function of the continuous parameters labeling the defect, the two-dimensional IR scales, and also of the Coulomb vev’s and the strong-coupling scale of the bulk gauge theory. The twisted superpotential thus describes the coupled 2d/4d system.
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k
1k
2. . .
k
M −1N
Figure 1. The quiver which describes the generic surface operator in pure SU(N ) gauge theory.
Surface operators in theories with eight supercharges can be studied from diverse points of view. One approach is to treat them as monodromy defects (also known as Gukov-Witten defects) in four dimensions along the lines discussed in [2,3] and compute the corresponding twisted superpotential W using equivariant localization as shown for example in [4–7]. A second approach is to focus on the two-dimensional world-volume theory on the surface operator [8]. In the superconformal theories of class S, a microscopic description of a generic co-dimension 4 surface operator in terms of (2, 2) supersymmetric quiver gauge theories in two dimensions was realized in [9]. Here we focus on a generic co-dimension 2 surface operator in pure N = 2 SU(N ) gauge theory [10], which has a microscopic description as a quiver gauge theory of the type shown in figure 1.
Here the round nodes, labeled by an index I, correspond to U(kI) gauge theories in two dimensions whose field strength is described by a twisted chiral field Σ(I). The rightmost node represents the four-dimensional N = 2 gauge theory whose SU(N ) gauge group acts as a flavor group for the last two-dimensional node. The arrows correspond to (bi-)fundamental matter multiplets that are generically massive. Integrating out these fields leads to an effective action for the twisted chiral fields which, because of the two-dimensional (2, 2)-supersymmetry, is encoded in a twisted chiral superpotential W . The contribution to W coming from the massive fields attached to the last node depends on the four-dimensional dynamics of the SU(N ) theory and in particular on its resolvent [10]. In this approach a key role is played by the twisted chiral ring equations that follow from extremizing the twisted superpotential with respect to Σ(I). The main idea is that by evaluating W on the solutions to the twisted chiral ring equations one should reproduce precisely the superpotential calculated using localization.
In this work, we extend this analysis in a few directions. We show that there exists a precise correspondence between the choice of massive vacua in two dimensions and the Gukov-Witten defects of the SU(N ) gauge theory labeled by the partition [n1, . . . nM] with n1+· · ·+nM = N . We also describe the relation between the (M −1) dynamically generated scales ΛI associated to the Fayet-Iliopoulos (FI) parameters for the two-dimensional nodes and the (M − 1) dimensionful parameters that naturally occur in the ramified instanton counting problem. Both the chiral ring equations and the localization methods can be extended to five-dimensional theories compactified on a circle of circumference β, i.e. to theories defined on R4× S1
β. In this case, surface operators correspond to codimension-2 defects wrapping Sβ1 and supporting a three-dimensional gauge theory coupled to the bulk five-dimensional theory. In the 3d/5d case, one again has a quiver theory, and as before its infrared dynamics is encoded in a twisted chiral superpotential. However, the form of the superpotential is modified to take into account the presence of a compactified direction.
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The twisted chiral rings for purely three-dimensional quiver theories have been studied ingreat detail in [11]. Here we extend this analysis and propose that the coupling between the last three-dimensional gauge node and the compactified five-dimensional theory is once again determined via the resolvent of the latter. With this assumption, the analysis of the modified twisted chiral ring equations as well as the choice of vacuum follow exactly the same pattern as in the 2d/4d case. An important and non-trivial check of this proposal is provided by the perfect agreement between the twisted superpotential obtained from solving the chiral ring equations and the one obtained from localization in five dimensions, which we perform in several examples.
In the 2d/4d case, the quiver theory on the defect can be mapped to other quiver theories by chains of Seiberg-like dualities, which lead to different quiver realizations of the same Gukov-Witten defect [7,12,13]. We show that, with an appropriate ansatz, the solutions of the twisted chiral ring equations for such dual theories lead to the same twisted superpotential. We obtain strong indications that each such superpotential matches the result of a localization computation carried out with a different residue prescription. If we promote the quiver theories to the compactified 3d/5d set-up, the superpotentials still agree at the classical level but, in general, they differ when quantum corrections are taken into account. The 3d/5d quiver gauge theories can be extended to include Chern-Simons (CS) terms in their effective action. Quite remarkably, we find in a simple but significant example that the equivalence between the dual quiver realizations of the same defect is restored at the quantum level if suitably chosen CS terms are added to the superpotentials.
The paper proceeds as follows. In section 2, we study the coupled 2d/4d system and solve the twisted chiral ring equations as power series in the IR scales of the theory. In section 3, we lift the discussion to coupled 3d/5d systems compactified on a circle. In section 4, we analyze the ramified instanton counting in four and five dimensions and show that the effective twisted chiral superpotential calculated using localization methods exactly matches the one obtained from the solution of the chiral ring equations in the earlier sections. In section 5, we discuss the relation between different quiver realizations of the same surface defect, and show that the equivalence between two dual realizations, which is manifest in the 2d/4d case, is in general no longer true in the 3d/5d case. We also show in a specific example that the duality is restored by adding suitable Chern-Simons terms. Finally, in section6we present our conclusions and discuss some possible extensions of our results. Some technical details are collected in the appendices.
2 Twisted superpotential for coupled 2d/4d theories
In this section our focus is the calculation of the low-energy effective action for surface operators in pure N = 2 SU(N ) supersymmetric gauge theories in four dimensions. As mentioned in the Introduction, surface operators can be efficiently described by means of a coupled 2d/4d system in which the two-dimensional part is a (2, 2)-supersymmetric quiver gauge theory with (bi-)fundamental matter, as shown in figure 1. Such coupled 2d/4d systems have an alternative description as Gukov-Witten monodromy defects [2,3]. The discrete data that label these defects correspond to the partitions of N , and can be
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summarized in the notation SU(N )[n1, . . . , nM] where n1+ · · · + nM = N . The M integersnI are related to the breaking pattern (or Levi decomposition) of the four-dimensional gauge group on the defect, namely
SU(N ) −→ SU(n1) × . . . × U(nM) . (2.1) They also determine the ranks kI of the two-dimensional gauge groups of the quiver in figure 1according to
kI = n1+ · · · + nI . (2.2)
The (bi-)fundamental fields connecting two nodes turn out to be massive. Integrating them out leads to the low-energy effective action for the gauge multiplet. In the I-th node the gauge multiplet is described by a twisted chiral field Σ(I) and the low-energy effective action is encoded in a twisted chiral superpotential W (Σ(I)). The vacuum structure can be determined by the twisted chiral ring equations, which take the form [14–16]
exp ∂W ∂Σ(I)s = 1 (2.3)
where Σ(I)s are the diagonal components, with s = 1, . . . , k [17]. This exponentiated form of the equations is a consequence of the electric fluxes which can be added to minimize the potential energy and which lead to linear (in Σ(I)) terms in the effective superpotential.
We extend this analysis in the following manner: first of all, we show that in the classical limit there is a very specific choice of solutions to the twisted chiral ring equations that allows us to make contact with the twisted chiral superpotential calculated using localization. We establish the correspondence between the continuous parameters labeling the monodromy defect and the dynamically generated scales of the two-dimensional quiver theory. We then show that quantum corrections in the quiver gauge theory are mapped directly to corrections in the twisted superpotential due to ramified instantons of the four-dimensional theory.
2.1 SU(2)
As an illustrative example, we consider the simple surface operator in the pure SU(2) theory which is represented by the partition [1, 1]. From the two-dimensional perspective, the effective dynamics is described by a non-linear sigma model with target-space CP1, coupled to the four-dimensional SU(2) gauge theory in a particular way that we now describe. We use the gauged linear sigma model (GLSM) description of this theory [17,18] in order to study its vacuum structure. We essentially follow the discussion in [10] although, as we shall see in detail, there are some differences in our analysis. The GLSM is a U(1) gauge theory with two chiral multiplets in the fundamental representation, that can be associated to the quiver drawn in figure 2.
Let us first analyze the simple case in which the quantum effects of the SU(2) theory are neglected. We consider a generic point in the Coulomb branch parameterized by the vev’s a1 = −a2 = a of the adjoint SU(2) scalar field Φ in the vector multiplet. These have the interpretation of twisted masses for the chiral multiplet of the two-dimensional
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1
2
Figure 2. The quiver representation of the SU(2)[1,1] surface operator.
U(1) gauge theory. The theory obtained by integrating out this massive multiplet has been studied in some detail in a number of works and here we merely present the resulting effective action which takes the simple form:1
W = 2πi τ (µ) σ − 2 X i=1 (σ − ai) logσ − ai µ − 1 = 2πi τ (µ) σ − Tr (σ − Φ) logσ − Φ µ − 1 . (2.4)
Here µ is the ultra-violet (UV) cut-off which we eventually take to infinity, and τ (µ) is the bare FI parameter at the scale µ. We can rewrite this superpotential using another scale µ0 and get W = 2πi τ (µ) − 2 logµ 0 µ σ − Tr (σ − Φ) logσ − Φ µ0 − 1 . (2.5)
From the coefficient of the linear term in σ, we identify the running of the FI coupling:2
2πi τ (µ0) = 2πi τ (µ) − 2 logµ 0
µ . (2.6)
In particular, we can choose to use the complexified IR scale Λ1at which τ (Λ1) = 0, so that W = −Tr (σ − Φ) logσ − Φ Λ1 − 1 . (2.7)
In this way we trade the UV coupling τ (µ) for the dynamically generated scale Λ1. Let us now turn on the dynamics of the four-dimensional SU(2) gauge theory. As pointed out in [10], this corresponds to considering the following superpotential:
W = − Tr (σ − Φ) logσ − Φ Λ1 − 1 . (2.8)
The angular brackets signify taking the quantum corrected vev of the chiral observable in the four-dimensional SU(2) theory. The twisted chiral ring equation is obtained by extremizing W and is given by
exp ∂W ∂σ
= 1 , (2.9)
1For notational simplicity we denote the superfield Σ by its lowest scalar component σ.
2Recall that τ is actually the complexification of the FI parameter r with the θ-angle: 2πiτ = iθ − r.
The sign of the coefficient of the logarithmic running (2.5) is such that r(µ0) grows with the scale µ0. The same is true in the other cases we consider.
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which, using the superpotential (2.8), is equivalent toexp Tr logσ − Φ Λ1 = 1 . (2.10)
As explained in [10], the left-hand side of (2.10) is simply the integral of the resolvent of the pure N = 2 SU(2) theory in four dimensions which takes the form [19]:
Tr logσ − Φ Λ1 = log P2(σ) +pP2(σ) 2− 4Λ4 2Λ21 ! . (2.11)
Here Λ is the four-dimensional strong coupling scale of the SU(2) theory and
P2(σ) = σ2− u (2.12)
is the characteristic polynomial appearing in the Seiberg-Witten solution where u = 1 2Tr Φ 2 = a2+ Λ4 2a2 + 5Λ8 32a6 + . . . (2.13)
Using (2.11) and performing some simple manipulations, we find that the twisted chiral ring relation (2.10) becomes
P2(σ) = Λ21+ Λ4 Λ2 1
(2.14) from which we obtain the two solutions
σ±?(u, Λ1) = ± s
u + Λ21+Λ 4
Λ21 . (2.15)
Notice the explicit presence of two different scales, Λ1 and Λ, which are related respec-tively to the two-dimensional and the four-dimensional dynamics. Clearly, the purely two-dimensional result can be recovered by taking the Λ → 0 limit. We can now substitute either one of the solutions of the chiral ring equation into the twisted chiral superpotential and obtain a function W?±. The proposal in [10] is that this should reproduce the twisted superpotential calculated using localization methods. We shall explicitly verify this in section 4, but here we would like to point out an important simplification that occurs in this calculation.
Let us consider the twisted effective superpotential evaluated on the σ+? solution of the chiral ring relations, namely
W?+ u, Λ1
≡ W σ?+(u, Λ1), Λ1 . (2.16) While W?+ itself is complicated, its logarithmic derivative with respect to Λ1 takes a re-markably simple form. In fact W?+ seems to depend on Λ1 both explicitly and through the solution σ?+, but on shell ∂W/∂σ
σ+? = 0 and so we simply have
Λ1 dW+ ? dΛ1 = Λ1 ∂W ∂Λ1 σ+? = 2σ?+ . (2.17)
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Using the explicit form of the solution given in (2.15), and inserting in it theweak-coupling expansion (2.13) of u, we thus obtain 1 2Λ1 dW?+ dΛ1 = a + 1 2a Λ21+Λ 4 Λ21 − 1 8a3 Λ41+Λ 8 Λ41 + 1 16a5 Λ61+ Λ21Λ4+Λ 8 Λ21 + Λ12 Λ61 + . . . . (2.18)
As we shall show later in section4, this result precisely matches the derivative of the twisted effective superpotential calculated using localization for the simple surface operator in the SU(2) gauge theory, provided we suitably relate the dynamically generated scale Λ1 of the two-dimensional theory to the ramified instanton counting parameter in presence of the monodromy defect.
2.2 Twisted chiral ring in quiver gauge theories
We will now show that the procedure described above generalizes in a rather simple way to any surface operator in the SU(N ) gauge theory labeled by a partition of N . In this case, however, it will not be possible to solve exactly the twisted chiral ring equations as we did in the SU(2) theory. We will have to develop a systematic perturbative approach in order to obtain a semi-classical expansion for the twisted chiral superpotential around a particular classical vacuum. Proceeding in this way we again find that the derivatives of the twisted superpotential with respect to the various scales have simple expressions in terms of combinations of the twisted chiral field σ evaluated in the appropriately chosen vacuum.
Following [10], we consider a quiver gauge theory of the form
U(k1) × U(k2) × . . . × U(kM −1) (2.19) with (bi)-fundamental matter between successive nodes, coupled to a pure N = 2 theory in four dimensions with gauge group SU(N ) acting as a flavor symmetry for the rightmost factor in (2.19). All this is represented in figure 1. We choose an ordering such that
k1 < k2< k3. . . < kM −1< N , (2.20) where the kI’s are related to the entries of the partition of N labeling the surface operator as indicated in (2.2). Our first goal is to obtain the twisted chiral ring of this 2d/4d system. Only the diagonal components of σ are relevant for this purpose [17], and thus for the I-th gauge group we take
σ(I) = diag σ1(I), σ2(I), . . . , σ(I)k
I . (2.21)
The (bi)-fundamental matter fields are massive and their (twisted) mass is proportional to the difference in the expectation values of the σ’s in the two nodes connected by the matter multiplet. In order to minimize the potential energy, the twisted chiral field σ(I) gets a vev and this in turn leads to a non-vanishing mass for the (bi)-fundamental matter.
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Integrating out these massive fields, we obtain the following effective superpotentialW = 2πi M −1 X I=1 kI X s=1 τI(µ) σs(I)− M −2 X I=1 kI X s=1 kI+1 X t=1 $ σs(I)− σt(I+1) − kM −1 X s=1 Tr $ σ(M −1)s − Φ (2.22)
where, for compactness, we have introduced the function $(x) = x logx µ− 1 (2.23)
with µ being the UV cut-off scale. Similarly to the SU(2) example previously considered, also here we can trade the UV parameters τI(µ) for the dynamically generated scales ΛI for each of the gauge groups in the quiver. To this aim, we unpackage of the terms containing the $-function and rewrite them as follows:
$ σs(I)− σt(I+1) = σ(I) s logσ (I) s − σt(I+1) ΛI − 1 − σt(I+1) logσ (I) s − σt(I+1) ΛI+1 − 1 + σs(I)logΛI µ − σ (I+1) t log ΛI+1 µ (2.24) for I = 1, . . . , M − 2, and Tr $ σs(M −1)− Φ = Tr " (σs(M −1)− Φ) logσ (M −1) s − Φ ΛM −1 − 1 # + N σ(M −1)s logΛM −1 µ . (2.25)
Considering the linear terms in the σ(I) fields we see that the FI couplings change with the scale and we can define the dynamically generated scales ΛI to be such that
τI(ΛI) = τI(µ) −
kI+1− kI−1 2πi log
ΛI
µ = 0 (2.26)
for I = 1, . . . , M − 1.3 Equivalently, we can write
ΛbI I = e
2πi τI(µ)µbI (2.27)
where
bI = kI+1− kI−1 (2.28)
denotes the coefficient of the β-function for the running of the FI parameter of the I-th node.
3We assume that k
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Using these expressions, the twisted superpotential (2.22) can thus be rewritten asW = − M −2 X I=1 kI X s=1 kI+1 X t=1 σs(I) logσ (I) s − σt(I+1) ΛI − 1 + M −1 X I=2 kI X s=1 kI−1 X r=1 σs(I) logσ (I−1) r − σ(I)s ΛI − 1 − kM −1 X s=1 * Tr " (σ(M −1)s − Φ) logσ (M −1) s − Φ ΛM −1 − 1 #+ . (2.29)
The I-th term (1 ≤ I ≤ M − 2) in the first line and the (I + 1)-th term in the second line of this expression are obtained by integrating out the bifundamental fields between the nodes I and (I + 1), while the last line is the result of integrating out the fundamental fields attached to the last gauge node of the quiver. The angular brackets account for the four-dimensional dynamics of the SU(N ) theory. One can easily verify that for N = M = 2, the expression in (2.29) reduces to (2.8).
The twisted chiral ring. The twisted chiral ring relations are given by exp ∂W
∂σs(I)
= 1 . (2.30)
In order to write the resulting equations in a compact form, we define a characteristic gauge polynomial for each of the SU(kI) node of the quiver
QI(z) = kI
Y
s=1
(z − σs(I)) . (2.31)
For I = 1, . . . , M − 2, the equations are independent of the four-dimensional theory, and read
QI+1(z) = (−1)kI−1ΛbIIQI−1(z) (2.32) with z = σs(I) for each s, and with the understanding that Q0 = 1 and k0 = 0. Note that the power of ΛI, which is determined by the running of the FI coupling, makes the equation consistent from a dimensional point of view. For I = M − 1, the presence of the four-dimensional SU(N ) gauge theory affects the last two-dimensional node of the quiver, and the corresponding chiral ring equation is
exp Tr log z − Φ ΛM −1 = (−1)kM −2ΛbM −1−N M −1 QM −2(z) (2.33)
with z = σs(M −1)for each s. We now use the fact that the resolvent of the four-dimensional SU(N ) theory, which captures all information about the chiral correlators, is given by [19]
T (z) := Tr 1 z − Φ = P 0 N(z) p PN(z)2− 4Λ2N (2.34)
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where PN(z) is the characteristic polynomial of degree N encoding the Coulomb vev’s of theSU(N ) theory and Λ is its dynamically generated scale. Since we are primarily interested in the semi-classical solution of the chiral ring equations, we exploit the fact that PN(z) can be written as a perturbation of the classical gauge polynomial in the following way:
PN(z) = N Y
i=1
(z − ei) (2.35)
where ei are the quantum vev’s of the pure SU(N ) theory given by [20,21]
ei = ai− Λ2N ∂ ∂ai Y j6=i 1 a2ij ! + O(Λ4N) . (2.36)
Integrating the resolvent (2.34) with respect to z and exponentiating the resulting expres-sion, one finds
exp Tr log z − Φ ΛM −1 = PN(z) + q PN(z)2− 4Λ2N 2ΛN M −1 . (2.37)
Using this, we can rewrite the twisted chiral ring relation (2.33) associated to the last node of the quiver in the following form:
PN(z) + q
PN(z)2− 4Λ2N = 2 (−1)kM −2ΛbM −1M −1QM −2(z) , (2.38) where z = σ(M −1)s . With further simple manipulations, we obtain
PN(z) = (−1)kM −2Λ bM −1 M −1QM −2(z) + Λ2N (−1)kM −2ΛbM −1 M −1QM −2(z) (2.39)
for z = σs(M −1). In the limit Λ → 0 which corresponds to turning off the four-dimensional dynamics, we obtain the expected twisted chiral ring relation of the last two-dimensional node of the quiver. Equations (2.32) and (2.39) are the relevant chiral relations which we are going to solve order by order in the ΛI’s to obtain the weak-coupling expansion of the twisted chiral superpotential.
Solving the chiral ring equations. Our goal is to provide a systematic procedure to solve the twisted chiral ring equations we have just derived and to find the effective twisted superpotential of the 2d/4d theory. As illustrated in the case of the SU(2) theory in section 2.1, we shall do so by evaluating W on the solutions of the twisted chiral ring equations. Each choice of vacuum therefore corresponds to a different surface operator.
In order to clarify this last point, we first solve the classical chiral ring equations, which are obtained by setting ΛI and Λ to zero keeping their ratio fixed, i.e. by considering the theory at a scale much bigger than ΛI and Λ. Thus, in this limit the right-hand sides
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of (2.32) and (2.39) vanish. A possible choice4 that accomplishes this is:σs(1)= as for s = 1, . . . , k1, σt(2)= at for t = 1, . . . , k2, .. . σw(M −1)= aw for w = 1, . . . , kM −1 . (2.40)
This is equivalent to assuming that the classical expectation value of σ for the I-th node is σ(I)= diag a1, a2, . . . , akI . (2.41)
We will see that this choice is the one appropriate to describe a surface defect that breaks the gauge group SU(N ) according to the Levi decomposition (2.1).
Let us now turn to the quantum chiral ring equations. Here we make an ansatz for σ(I) as a power series in the various Λ
I’s around the chosen classical vacuum. From the explicit expressions (2.32) and (2.39) of the chiral ring equations, it is easy to realize that there is a natural set of parameters in terms of which these power series can be written; they are given by
qI = (−1)kI−1ΛbII (2.42)
for I = 1, . . . , M − 1. If the four-dimensional theory were not dynamical, these (M − 1) parameters would be sufficient; however, from the chiral ring equations (2.39) of the last two-dimensional node of the quiver, we see that another parameter is needed. It is related to the four-dimensional scale Λ and hence to the four-dimensional instanton action. It turns out that this remaining expansion parameter is
qM = (−1)NΛ2N M −1 Y I=1 qI !−1 . (2.43)
Our proposal is to solve the chiral ring equations (2.32) and (2.39) as a simultaneous power series in all the qI’s, including qM, which ultimately will be identified with the Nekrasov-like counting parameters in the ramified instanton computations described in section 4.
We will explicitly illustrate these ideas in some examples in the next section, but first we would like to show in full generality that the logarithmic derivatives with respect to ΛI are directly related to the solution σ?(I)of the twisted chiral ring equations (2.32) and (2.39). The argument is a straightforward generalization of what we have already seen in the SU(2) case, see (2.16) and (2.17). On shell, i.e. when ∂W/∂σ
σ? = 0, the twisted superpotential
W? ≡ W (σ?) depends on ΛI only explicitly. Using the expression of W given in (2.29), we find ΛI dW? dΛI = ΛI ∂W ∂ΛI σ? = bItr σ?(I), (2.44) where in the last step we used the tracelessness of Φ. This relation can be written in terms of the parameters qI defined in (2.42), as follows
qI dW?
dqI
= tr σ(I)? . (2.45)
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1
3
Figure 3. The quiver corresponding to the surface operator SU(3)[1,2].
If we express the solution σ? of the chiral ring equations as the classical solution (2.40) plus quantum corrections, we find
q1 dW? dq1 = a1+ . . . ak1+ corr.ns = a1+ . . . + an1 + corr.ns , q2 dW? dq2 = a1+ . . . ak2+ corr.ns = a1+ . . . + an1+n2 + corr.ns (2.46)
and so on. This corresponds to a partition of the classical vev’s of the SU(N ) theory given by n | {z } n1 a1, · · · an1, | {z } n2 an1+1, · · · an1+n2, · · · , | {z } nM aN −nM+1, . . . aN o , (2.47)
which is interpreted as a breaking of the gauge group SU(N ) according to the Levi de-composition (2.1). In fact, by comparing with the results of [6] (see for instance, equation (4.1) of this reference), we see that the expressions (2.46) coincide with the derivatives of the classical superpotential describing the surface operator of the SU(N ) theory, labeled by the partition [n1, n2, . . . , nM], provided we relate the parameters qI to the variables tJ that label the monodromy defect according to
2πi tJ ∼ M X
I=I
log qI for J = 1, . . . , M . (2.48)
We now illustrate these general ideas in a few examples. 2.3 SU(3)
We consider the surface operators in the SU(3) theory. There are two distinct partitions, namely [1, 2] and [1, 1, 1], which we now discuss in detail.
SU(3)[1,2]. In this case the two-dimensional theory is a U(1) gauge theory with three flavors, represented by the quiver in figure 3.
Since M = 2, we have just one σ and one chiral ring equation, which is given by (see (2.39))
P3(σ) = Λ31+ Λ6
Λ31 (2.49)
where the gauge polynomial is defined in (2.35). We solve this equation order by order in Λ1 and Λ, using the ansatz
σ? = a1+ X `1,`2 c`1,`2q `1 1 q `2 2 (2.50)
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1
2
3
Figure 4. The quiver diagram representing the surface operator SU(3)[1,1,1].
where the expansion parameters are defined in (2.42) and (2.43), which in this case explic-itly read q1= Λ31, q2 = − Λ6 Λ3 1 . (2.51)
Inserting (2.50) into (2.49), we can recursively determine the coefficients c`1`2 and, at the
first orders, find the following result σ?= a1+ 1 a12a13 Λ31+Λ 6 Λ31 − 1 a312a213 + 1 a212a313 Λ61+Λ 12 Λ61 + . . . (2.52) where aij = ai− aj. According to (2.45), this solution coincides with the q1-logarithmic derivative of the twisted superpotential. We will verify this statement by comparing (2.52) against the result obtained via localization methods.
SU(3)[1,1,1]. In this case the two-dimensional theory is represented by the quiver of figure 4.
Since M = 3, there are now two sets of twisted chiral ring equations. For the first node, from (2.32) we find
2 Y
s=1
σ(1)− σ(2)
s = Λ21, (2.53)
while for the second node, from (2.39) we get P3 σs(2) = −Λ22 σ(2)s − σ(1) −
Λ6 Λ22 σ(2)s − σ(1)
(2.54)
for s = 1, 2. From the classical solution to these equations (see (2.41)), we realize that this configuration corresponds to a surface operator specified by the partition of the Coulomb vev’s {{a1}, {a2}, {a3}}, which is indeed associated to the partition [1,1,1] we are consid-ering. Thus, the ansatz for solving the quantum equations (2.53) and (2.54) takes the following form: σ?(1)= a1+ X `1,`2,`3 d`1,`2,`3 q `1 1 q `2 2 q `3 3 , σ?,1(2)= a1+ X `1,`2,`3 f`1,`2,`3 q `1 1 q `2 2 q `3 3 , σ?,2(2)= a2+ X `1,`2,`3 g`1,`2,`3 q `1 1 q `2 2 q `3 3 , (2.55) with q1 = Λ21, q2 = −Λ22, q3= Λ6 Λ21Λ22 . (2.56)
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Solving the coupled equations (2.53) and (2.54) order by order in qI, we find the followingresult: σ?(1)= a1+ 1 a12 Λ21+ 1 a13 Λ6 Λ21Λ22 − 1 a312Λ 4 1− 1 a313 Λ12 Λ41Λ42 − 1 a12a13a23 Λ21Λ22−Λ 6 Λ2 1 + . . . , (2.57) Tr σ?(2)= a1+ a2− 1 a23 Λ22+ 1 a13 Λ6 Λ2 1Λ22 − 1 a3 23 Λ42− 1 a3 13 Λ12 Λ4 1Λ42 − 1 a12a13a23 Λ21Λ22+ Λ 6 Λ2 2 + . . . . (2.58)
According to (2.45) these expressions should be identified, respectively, with the q1- and q2-logarithmic derivatives of the twisted superpotential. We will verify this relation in section 4using localization.
We have analyzed in detail the SU(3) theory in order to exhibit how explicit and systematic our methods are. We have thoroughly explored all surface defects in the SU(4) and SU(5) theories and also considered a few other examples with higher rank gauge groups. In all these cases our method of solving the twisted chiral ring equations proved to be very efficient and quickly led to very explicit results. One important feature of our approach is the choice of classical extrema of the twisted superpotential which will allow us to make direct contact with the localization calculations of the superpotential for Gukov-Witten defects in four-dimensional gauge theories. A further essential ingredient is the use of the quantum corrected resolvent in four dimensions, which plays a crucial role in obtaining the higher-order solutions of the twisted chiral ring equations of the two-dimensional quiver theory.
3 Twisted superpotential for coupled 3d/5d theories
Let us now consider the situation in which the 2d/4d theories described in the previous section are replaced by 3d/5d ones compactified on a circle Sβ1 of length β. The content of these theories is still described by quivers of the same form as in figure 1. We begin by considering the three-dimensional part.
3.1 Twisted chiral ring in quiver gauge theories
To construct the effective theory for the massless chiral twisted fields, which is encoded in the twisted superpotential W , we have to include the contributions of all Kaluza-Klein (KK) copies of the (bi)-fundamental matter multiplets. When the scalars σ(I) are gauge-fixed as in (2.21), the KK copies of the matter multiplets have masses5
σs(I)− σt(I+1)+ 2πi n/β , (3.1)
5This is consistent with the fact that gauge-fixing the scalars σ(I)as in (2.21) leaves a residual invariance
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for I = 1, . . . , M − 2 (with an independent integer n for each multiplet). Similarly, thecopies of the matter multiplet attached to the 5d node have masses
σs(M −1)− ai+ 2πi n/β (3.2)
when the 5d theory is treated classically.
All these chiral massive fields contribute to the one-loop part of W . As we saw in (2.22) and (2.23), in 2d a chiral field of mass z contributes a term proportional to $(z). Summing over all its KK copies results therefore in a contribution proportional to
`(z) ≡X n∈Z
$ z + 2πi n/β (3.3)
where the sum has to be suitably regularized. This function satisfies the property ∂z`(z) = X n∈Z ∂z$ z + 2πi n/β = X n∈Z log z + 2πi n/β µ = log 2 sinhβz 2 . (3.4) Note that the scale µ, present in the definition (2.23) of the function $(z), no longer appears after the sum over the KK copies. Integrating this relation, one gets
`(z) = 1 β Li2 e −βz + βz2 4 − π2 6β , (3.5)
where the integration constant has been fixed in such a way that `(z) β→0∼ z
log(βz) − 1
= $(z) . (3.6)
Note that here $(z) is defined taking the UV scale to be
µ = 1/β , (3.7)
as is natural in this compactified situation.
Therefore, the twisted superpotential of the three-dimensional theory is simply given by (2.22) with all occurrences of the function $(z) replaced by `(z), for any argument z, and with the UV scale µ being set to 1/β. Just as in the two-dimensional case, we would like to replace the FI couplings at the UV scale, τI(1/β), with the dynamically generated scales ΛI. Since the renormalization of these FI couplings is determined only by the lightest KK multiplets, the running is the same as in two dimensions and thus we can simply use (2.26) with µ identified with 1/β according to (3.7). Note however that, in contrast to the two-dimensional case described in (2.29), this replacement does not eliminate completely the UV scale from the expression of W , and the dependence on β remains in the functions `(z). Altogether we have W = M −1 X I=1 kI X s=1 bIlog(βΛI)σ(I)s − M −2 X I=1 kI X s=1 kI+1 X t=1 ` σ(I)s − σ(I+1)t − kM −1 X s=1 Tr ` σs(M −1)− Φ . (3.8) The expectation value in the last term is taken with respect to the five-dimensional gauge theory defined on the last node of the quiver and compactified on the same circle of length β as the three-dimensional sector.
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The twisted chiral ring. Our aim is to show that the twisted superpotential (3.8),evaluated on a suitably chosen vacuum σ?, matches the twisted superpotential extracted via localization for a corresponding monodromy defect. Just as in the 2d/4d case, the vacuum σ? minimizes W , namely solves the twisted chiral ring equation (2.30). Moreover, the logarithmic derivatives of W with respect to ΛI, or with respect to the parameters qI in (2.42), evaluated on a solution σ?, still satisfy (2.44) or (2.45) respectively. These derivatives are quite simple to compute and these are the quantities that we will compare with localization results.
In close parallel to what we did in the 2d/4d case, the chiral ring equations (2.30) can be expressed in a compact form if we introduce the quantity
b QI(z) = kI Y s=1 2 sinhβ(z − σ (I) s ) 2 (3.9)
for each of the SU(kI) gauge groups in the quiver; note that bQI(z) is naturally written in terms of the exponential variables
Ss(I)= eβσ(I)s , (3.10)
which are invariant under the shifts described in footnote 5. Indeed, starting from (2.30) and taking into account (3.4), for I = 1, . . . , M − 2 we find
b
QI+1(z) = (−1)kI−1 βΛI bI
b
QI−1(z) (3.11)
with z = σ(I)s . For the node I = M − 1 we obtain exp Tr log 2 sinhβ(z − Φ) 2 = (−1)kM −2 βΛ M −1 bM −1 b QM −2(z) (3.12)
with z = σs(M −1). To proceed further, we need to evaluate in the compactified 5d theory the expectation value appearing in the left hand side of (3.12). To do so, let us briefly recall a few facts about this compactified gauge theory.
The resolvent in the compactified 5d gauge theory. The five-dimensional N = 1 vector multiplet consists of a gauge field Aµ, a real scalar φ and a gluino λ. Upon circle compactification, the component Atof the gauge field along the circle and the scalar φ give rise to the complex adjoint scalar Φ = At+ iφ of the four-dimensional N = 2 theory. The Coulomb branch of this theory is classically specified by fixing the gauge [22]:
Φ = At+ iφ = diag (a1, a2, . . . aN) . (3.13) However, there is a residual gauge symmetry under which
ai → ai+ 2πi ni/β (3.14)
with ni ∈ Z; since we are considering a SU(N) theory, we must ensure that these shifts preserve the vanishing of P
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The low-energy effective action can be determined in terms of an algebraic curve anda differential, just as in the usual four-dimensional case. The Seiberg-Witten curve for this model was first proposed in [22] and later derived from a saddle point analysis of the instanton partition function in [23,24]; it takes the following form
y2 = bPN2(z) − 4 βΛ2N
. (3.15)
Here Λ is the strong-coupling scale that is dynamically generated and
b PN(z) = N Y i=1 2 sinhβ(z − ei) 2 (3.16) where ei parametrize the quantum moduli space and reduce to ai in the classical regime, in analogy to the four-dimensional case. Like the latter, they also satisfy a tracelessness con-dition: P
iei= 0. Note that bPN can be written purely in terms of the exponential variables
Ei= eβei, Z = eβz, (3.17)
and is thus invariant under the shift (3.14). Indeed, using (3.17) we find
b PN(z) = Z− N 2 ZN + N −1 X i=1 (−1)iZN −iUi+ (−1)N , (3.18)
where Ui is the symmetric polynomial Ui =
X
j1<j2<...ji
Ej1. . . Ejk . (3.19)
In (3.18) we have used the SU(N ) tracelessness condition, which implies UN = Q
iEi = 1. The resolvent of this five-dimensional theory, defined as [25]
b T (z) = Tr cothβ(z − Φ) 2 = 2 β ∂ ∂z Tr log 2 sinhβ(z − Φ) 2 , (3.20) contains the information about the chiral correlators through the expansion
b T (z) = N + 2 ∞ X `=1 e−`βz Tr e` βΦ . (3.21)
On the other hand, the Seiberg-Witten theory expresses this resolvent as6
b T (z) = 2 β b PN0 (z) q b PN2(z) − 4 (βΛ)2N , (3.22)
so that, integrating (3.20), we have
exp Tr log 2 sinhβ(z − Φ) 2 = b PN(z) + q b PN2(z) − 4(βΛ)2N 2 . (3.23)
6Using (3.18) we can expand this expression in inverse powers of Z; then, comparing to (3.21), we can
relate the correlators hTr e` βΦi, of which the first (N − 1) ones are independent, to the U `’s.
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With manipulations very similar to those described in section 2for the 2d/4d case, we cannow rewrite the twisted chiral ring relation (3.12) as follows b PN(z) = (−1)kM −2 βΛM −1 bM −1 b QM −2(z) + βΛ 2N (−1)kM −2 βΛ M −1 bM −1 b QM −2(z) (3.24)
for z = σ(M −1)s . It is easy to check that in the limit β → 0 we recover the corresponding equation (2.39) for the 2d/4d theory.
Solving the chiral ring equations. At the classical level the solution to the chiral ring equations takes exactly the same form as in (2.41). In terms of the exponential variables introduced in (3.10) we can write it as
S?(I) = diag(A1, . . . , AkI) (3.25)
where Ai= eβai. These variables Ai represent the classical limit of the variables Ei defined in (3.17). The SU(N ) tracelessness condition implies that Q
iAi = 1.
Our aim is to solve the chiral ring equations (3.11) and (3.24), and then compare the solutions to the localizations results, which naturally arise in a semi-classical expansion. Therefore, we propose an ansatz that takes the form of an expansion in powers of β, namely
S?(I) = diag A1+ X ` δS1,`(I), . . . , AkI + X ` δSk(I) I,` . (3.26)
Notice that also the chiral ring equation (3.24) of the last node can be expanded in β. Indeed, the quantity bPN contains the moduli space coordinates Ui, which as shown in appendixA, admit a natural expansion in powers of βΛ2N. Putting everything together, we can solve all chiral ring equations iteratively, order by order in β and determine the corrections δSs,`(I) and thus the solution S?(I). In this way, repeating the same steps of the 2d/4d theories, we obtain the expression of the logarithmic derivative of the twisted superpotential, namely qI dW? dqI = 1 β kI X s=1
log S?,s(I) = Tr σ(I)? . (3.27)
3.2 SU(2) and SU(3)
We now show how this procedure works in a few simple examples with gauge groups of low rank.
SU(2)[1,1]. In this case the quiver is the one drawn in figure 2. Since M = 2, there is a single variable σ for the U(1) node and a single FI parameter τ . The only chiral ring equation is given by (3.24) with z = σ, namely
b P2(σ) = β2 Λ21+Λ 4 Λ21 . (3.28)
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Using (3.18) we can express bP2 in terms of S = eβσ, obtainingb
P2(S) = S + 1
S − U1 = 2 cosh(βσ) − U1 (3.29) where U1 = E1+ E2. A solution of the twisted chiral ring equation is therefore given by
σ? = 1 β log S? = 1 βarccosh U1 2 + β2 2 Λ21+ Λ 4 Λ2 1 . (3.30)
In appendix A we derive the semi-classical expansion of U1. This is given in (A.4) which, rewritten in terms of a, reads
U1 = 2 cosh(βa) 1 + βΛ 4 4 sinh2(βa) + . . . . (3.31)
Substituting this into (3.30), we find finally σ? = a + β 2 sinh(βa) Λ21+Λ 4 Λ21 −β 3cosh(βa) 8 sinh3(βa) Λ41+Λ 8 Λ41 + . . . . (3.32) According to (3.27), this solution corresponds to the logarithmic q1-derivative of the su-perpotential, namely
q1 dW?
dq1
= σ? . (3.33)
We will verify in the next section that this is indeed the case, by comparing with the superpotential computed via localization and finding a perfect match.
SU(3)[1,2]. This case is described by the quiver in figure 3. Again, we have M = 2 and thus a single variable σ and a single FI parameter τ . In this case, the chiral ring equation (3.24) reads b P3(S) = β3 Λ31+ Λ 6 Λ3 1 (3.34) where b P3(S) = S−3/2 S3− U1S2+ U2S − 1 . (3.35) Using the semi-classical expansions of U1 and U2 given in (A.5) and (A.6), and solving the chiral ring equation order by order in β according to the ansatz (3.26), we obtain
σ?= 1 β log S? = a1+ β 2 A 1/2 1 A12A13 Λ31+Λ 6 Λ31 − β 5 2 A1(A1+ A2) A312A213 + A1(A1+ A3) A212A313 Λ61+Λ 12 Λ61 + . . . (3.36)
where Aij = Ai− Aj. Rewriting this solution in terms of the classical vev’s ai, we have σ? = a1+ β2 4 sinh β2a12 sinh β2a13 Λ31+Λ 6 Λ31 (3.37) −β 5 32 cosh β2a12 sinh3 β2a12 sinh2 β2a13 + cosh β2a13 sinh2 β2a12 sinh3 β2a13 Λ61+Λ 12 Λ6 1 + . . . It is very easy to see that in the limit β → 0 this reduces to the solution of the corresponding 2d/4d theory given in (2.52). In the next section we will recover this same result by computing the q1-logarithmic derivative of the twisted superpotential using localization.
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SU(3)[1,1,1]. In this case the quiver is the one drawn in figure4. Since M = 3, we havetwo FI parameters and two sets of chiral ring equations. For the first node the equation is given by (3.11) which, in terms of the exponential variables, explicitly reads
2 Y s=1 (S(1)− S(2) s ) = β2S(1) q S1(2)S2(2)Λ21 . (3.38)
For the last node, instead, the chiral ring equations are given by (3.24), namely
b P3 Ss(2) = −β2 Λ22 S (2) s − S(1) q S(1)S(2) s +β 2Λ6 Λ2 2 q S(1)S(2)s Ss(2)− S(1) (3.39)
for s = 1, 2. Here bP3 is as in (3.35) with U1 and U2 given in (A.5) and (A.6). Solving these equations by means of the ansatz (3.26), we obtain
σ(1)? = 1 β log S (1) ? = a1+ β √ A1A2 A12 Λ21+ β √ A1A3 A13 Λ6 Λ21Λ22 + . . . , (3.40) and Tr σ?(2)= 1 β log S?,1(2)+ log S?,2(2) = a1+ a2− β √ A2A3 A23 Λ22+ β √ A1A3 A13 Λ6 Λ21Λ22 + . . . . (3.41)
In terms of the classical vev’s ai these solutions become, respectively, σ?(1)= a1+ β 2 sinh β2a12 Λ 2 1+ β 2 sinh β2a13 Λ6 Λ21Λ22 + . . . , (3.42) and Tr σ(2)? = a1+ a2− β 2 sinh β2a23 Λ 2 2+ β 2 sinh β2a13 Λ6 Λ2 1Λ22 + . . . . (3.43) In the limit β → 0 these expressions reproduce the first few terms of the 2d/4d solu-tions (2.57) and (2.58) and, as we will see in the next section, they perfectly agree with the qI-logarithmic derivatives of the twisted superpotential calculated using localization, confirming (3.27).
We have also computed and checked higher order terms in these SU(3) examples, as well as in theories with gauge groups of higher rank (up to SU(6)).
4 Ramified instantons in 4d and 5d
In this section we treat the surface operators as monodromy defects D. We begin by considering the four-dimensional case and later we will discuss the extension to a five-dimensional theory compactified on a circle of length β.
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4.1 Localization in 4dWe parametrize R4 ' C2 by two complex variables (z
1, z2) and place D at z2 = 0, filling the z1 plane. The presence of the surface operator induces a singular behavior in the gauge connection A, which acquires the following generic form [4,26]:
A = Aµdxµ ' − diag | {z } n1 γ1, · · · , γ1, | {z } n2 γ2, · · · , γ2, · · · , | {z } nM γM, · · · , γM dθ (4.1)
as r → 0. Here (r, θ) denote the polar coordinates in the z2-plane orthogonal to D, and γI are constant parameters that label the surface operator. The M integers nI are a partition of N and identify a vector ~n associated to the symmetry breaking pattern of the Levi decomposition (2.1) of SU(N ). This vector also determines the split of the vev’s ai according to (2.47).
A detailed derivation of the localization results for a generic surface operator has been given in [4–6], following earlier mathematical work in [27–29]. Here, we follow the discussion in [6] to which we refer for details, and present merely those results that are relevant for the pure gauge theory. The instanton partition function for a surface operator described by ~n is given by7 Zinst[~n] = X {dI} Z{dI}[~n] with Z{dI}[~n] = M Y I=1 (−qI)dI dI! Z dI Y σ=1 dχI,σ 2πi z{dI} (4.2) where z{dI} = M Y I=1 dI Y σ,τ =1 g (χI,σ− χI,τ + δσ,τ) g (χI,σ − χI,τ+ 1) × M Y I=1 dI Y σ=1 dI+1 Y ρ=1 g (χI,σ− χI+1,ρ+ 1+ ˆ2) g (χI,σ− χI+1,ρ+ ˆ2) (4.3) × M Y I=1 dI Y σ=1 nI Y s=1 1 g aI,s− χI,σ+12(1+ ˆ2) nI+1 Y t=1 1 g χI,σ− aI+1,t+12(1+ ˆ2) . Let us now explain the notation. The M positive integers dI count the numbers of ramified instantons in the various sectors, with the convention that dM +1 = d1.8 When these numbers are all zero, we understand that Z{dI=0}= 1. The M variables qI are the ramified
instanton weights, which will be later identified with the quantities qI used in the previous sections (see in particular (2.42) and (2.43)). The parameters 1 and ˆ2 = 2/M specify the Ω-background [23, 24] which is introduced to localize the integrals over the instanton moduli space; the rescaling by a factor of M in 2 is due to the ZM-orbifold that is used in the ramified instanton case [4]. Finally, the function g is simply
g(x) = x . (4.4)
This seems an unnecessary redundancy but we have preferred to introduce it because, as we will see later, in the five-dimensional theory the integrand of the ramified instanton
7Here, differently from [6], we have introduced a minus sign in front of q
I in order to be consistent with
the conventions chosen in the twisted chiral ring.
8Also in n
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partition function will have exactly the same form as in (4.3), with simply a differentfunction g.
The integrations over χI in (4.2) have to be suitably defined and regularized, and we will describe this in detail. But first we discuss a few consequences of the integral expression itself and show how to extract the twisted chiral superpotential from Zinst.
An immediate feature of (4.3) is that, unlike the case of the N = 2? theory studied in [6], the counting parameters qI have a mass dimension. In order to fix it, let us consider the contribution to the partition function coming from the one-instanton sector. This is a sum over M terms, each of which has dI = 1 for I = 1, . . . , M . Explicitly, we have
Z1−inst= − M X I=1 Z dχ I 2πi qI 1 nI Y s=1 1 aI,s− χI +12(1+ ˆ2) nI+1 Y t=1 1 χI − aI+1,t+12(1+ ˆ2) . (4.5) Since the partition function is dimensionless and χI carries the dimension of a mass, we deduce that mass dimension of qI is
qI = nI + nI+1= bI (4.6)
where the last step follows from combining (2.2) and (2.28). Another important dimensional constraint follows once we extract the non-perturbative contributions to the prepotential F and to the twisted effective superpotential W from Zinst. This is done by taking the limit in which the Ω-deformation parameters i are set to zero according to [4,26,30]
log Zinst= − Finst 1ˆ2 +Winst 1 + . . . (4.7)
where the ellipses refer to regular terms. The key point is that the prepotential extracted this way depends only on the product of all the qI. On the other hand, it is well-known that the instanton contributions to the prepotential are organized at weak coupling as a power series expansion in Λ2N where Λ is the dynamically generated scale of the four-dimensional theory and 2N is the one-loop coefficient of the gauge coupling β-function. Thus, we are naturally led to write9
M Y
I=1
qI = (−1)NΛ2N . (4.8)
Notice that the mass-dimensions (4.6) attributed to each of the qI are perfectly consistent with this relation, since the integers nI form a partition of N . We therefore find that we can use exactly the same parametrization used in the effective field theory and given in (2.42) and (2.43), which we rewrite here for convenience
qI = (−1)kI−1 ΛbII for I = 1, . . . , M − 1 , qM = (−1)NΛ2N M −1 Y I=1 qI −1 . (4.9)
9The sign in this formula is the one that, given our conventions, is consistent with the standard field
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Weight Poles Y ZY q1 χ1,1= a +12(1+ ˆ2) ( , •) 1 1(2a+1+ˆ2) q2 χ2,1= −a +12(1+ ˆ2) (•, ) 1 1(−2a+1+ˆ2) q1q2 χ1,1= a +12(1+ ˆ2) χ2,1= −a +12(1+ ˆ2) ( , ) − 1 21(4a2−ˆ22) q1q2 χ1,1= a +12(1+ ˆ2) χ2,1= χ1,1+ ˆ2 ( , •) − 1 21ˆ2(2a+ˆ2)(2a+1+ˆ2) q1q2 χ2,1= −a +12(1+ ˆ2) χ1,1= χ2,1+ ˆ2 (•, ) − 1 21ˆ2(ˆ2−2a)(−2a+1+ˆ2) q2 1 χ1,1= a +12(1+ ˆ2) χ1,2= χ1,1+ 1 , • 1 22 1(2a+1+ˆ2)(2a+21+ˆ2) q22 χ2,1= −a +12(1+ ˆ2) χ2,2= χ2,1+ 1 •, 1 22 1(−2a+1+ˆ2)(−2a+21+ˆ2)Table 1. We list the weight factors, the locations of the poles, the corresponding Young diagrams, and the contribution to the partition function in all cases up to two boxes for the SU(2) theory. Here we have set a1= −a2= a.
Residues and contour prescriptions. The last ingredient we have to specify is how to evaluate the integrals over χI in (4.2). The standard prescription [6,31–33] is to consider aI,s to be real and then close the integration contours in the upper-half χI,σ-planes with the choice
Im ˆ2 Im 1 > 0 . (4.10)
It is by now well-established that with this prescription the multi-dimensional integrals receive contributions from a subset of poles of z{dI}, which are in one-to-one correspondence
with a set of Young diagrams Y = {YI,s}, with I = 1, · · · , M and s = 1, · · · nI. This fact can be exploited to organize the result in a systematic way (see for example [6] for details). Let us briefly illustrate this for SU(2), for which there is only one allowed partition, namely [1, 1], and hence one single surface operator to consider [34]. In table1 we list the explicit results for this case, including the location of the poles and the contribution due to all the relevant Young tableaux configurations up to two boxes.
Combining these results, we find that the instanton partition function takes the fol-lowing form Zinst[1, 1] = 1 + q1 1(2a + 1+ ˆ2) + q2 1(−2a + 1+ ˆ2) + q 2 1 22 1(2a + 1+ ˆ2) (2a + 21+ ˆ2) + q 2 2 22 1(−2a + 1+ ˆ2) (−2a + 21+ ˆ2) + q1q2 1+ ˆ2 21ˆ2(−2a + 1+ ˆ2) (2a + 1+ ˆ2) + . . . (4.11)
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The prepotential and the twisted effective superpotential are extracted according to (4.7)and using the map (4.9). Let us focus on the twisted superpotential Winst, or better on its q1-derivative. We find q1 dWinst dq1 = 1 2a Λ21 +Λ 4 Λ21 − 1 8a3 Λ41+Λ 8 Λ41 + . . . . (4.12)
This precisely matches, up to two instantons, the non-perturbative part of the result (2.18) obtained by solving the twisted chiral ring equations for the quiver theory representing the surface defect in SU(2). We have also checked the agreement at higher instanton orders (up to six boxes), which we have not reported here for brevity.
The specific prescription (4.10) we have chosen to compute the instanton partition function is particularly nice due to the correspondence of the residues with Young tableaux. However, there are many other possible choices of contours that one can make. One way to classify these distinct contours is using the Jeffrey-Kirwan (JK) prescription [35]. In this terminology, the set of poles chosen to compute the residues is described by a JK parameter η, which is a particular linear combination of the χI,s; the prescription chooses a set of factors D from the denominator of z{dI} such that, if we only consider the χI,s-dependent
terms of these chosen factors, then, η can be written as a positive linear combination of these. For instance, our prescription in (4.10) corresponds to choosing10
η = − M X
I=1
χI (4.13)
For a detailed discussion of this method in the context of ramified instantons we refer to [7] where it is also shown that different JK prescriptions can be mapped to different quiver realizations of the surface operator.
Let us consider for example the prescription corresponding to a JK parameter of the form η = − M −1 X I=1 χI + ζ χM (4.14)
where ζ is a large positive number. In our notation this corresponds to closing the inte-gration contours in the upper half-plane as before for the first (M − 1) variables, and in the lower half plane for χM. Applying this new prescription to the SU(2) theory, we find a different set of poles that contribute. They are explicitly listed in table 2.
Comparing with table 1, we see that, although the location of residues has changed, for most cases the residues are unchanged. The only set of residues that give an apparently different answer is the one with d1 = d2 = 1 with weight q1q2. As opposed to the earlier case, where there were three contributions, now there are only two terms proportional to q1q2. However, it is easy to see that if we sum these contributions, we find an exact match between the two prescriptions. This fact should not come as surprise since it is a simple consequence of the residue theorem applied to the χ2 integral. Therefore, all
10We understand the extra index s running from 1 to n I.
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weight poles ZY q1 χ1,1= a + 12(1+ ˆ2) 1(2a+11+ˆ2) q2 χ2,1= a − 12(1+ ˆ2) 1(−2a+1 1+ˆ2) q1q2 χ1,1 = χ2,1+ ˆ2 χ2,1= −a − 12(1+ 3ˆ2) −2 1 1ˆ2(2a+ˆ2)(2a+1+ˆ2) q1q2 χ1,1 = χ2,1+ ˆ2 χ2,1 = a −12(1+ ˆ2) 1+2ˆ2 22 1ˆ2(2a+ˆ2)(−2a+1+ˆ2) q12 χ1,1 = a1+12(1+ ˆ2) χ1,2 = χ1,1+ 1 1 22 1(2a+1+ˆ2)(2a+21+ˆ2) q22 χ2,1= a − 12(1+ ˆ2) χ2,2 = χ2,1− 1 1 22 1(−2a+1+ˆ2)(−2a+21+ˆ2)Table 2. We list the weight factors, the pole structure and the contribution to the partition function in all cases up to two boxes for the SU(2) theory using the contour prescription corresponding to the JK parameter (4.14).
results that follow from the instanton partition function (and in particular the twisted superpotential) are the same in the two cases. Of course what we have just seen in the simple SU(2) case at the two instanton level, occurs also at higher instanton numbers and with higher rank gauge groups. The price one pays in changing the contour prescription or equivalently in changing the JK parameter from (4.13) to (4.14) is the loss of a simple one-to-one correspondence with the Young tableaux, but the gain is that, as shown in [7], the second prescription produces at each instanton order an instanton partition that is already organized in a factorized form in which the various factors account for the 2d, the 4d and the mixed 2d/4d contributions. This is a feature that will play a fundamental role in the 3d/5d extension.
Let us now list our findings obtained by using the second residue prescription for the SU(3) theory, limiting ourselves to the one-instanton terms for brevity. In the case of the surface operator corresponding to the partition [1,2] we get
Zinst[1, 2] = 1 + q1 1(a12+ 1+ ˆ2) (a13+ 1+ ˆ2) + q2 1(a21+ 1+ ˆ2) (a31+ 1+ ˆ2) + . . . (4.15) while for the surface operator described by the partition [1,1,1] we obtain
Zinst[1, 1, 1] = 1 + q1 1(a12+ 1+ ˆ2) + q2 1(a23+ 1+ ˆ2) + q3 1(a31+ 1+ ˆ2) + . . . . (4.16)
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Applying (4.7) to extract Winst, we find that the qI-logarithmic derivatives of the twistedsuperpotential for the two partitions perfectly match the non-perturbative pieces of the solutions (2.57) and (2.58) of the twisted chiral ring equations. We have checked that this agreement persists at the two-instanton level. We have also thoroughly explored all surface operators in the SU(4) theory and many cases in higher rank theories up to two instantons, always finding a perfect match between the qI-logarithmic derivatives of W and the solutions of the corresponding twisted chiral ring equations.
4.2 Localization in 5d
We now turn to discuss the results for a gauge theory on R4× S1
β in the presence of a surface operator also wrapping the compactification circle. This case has been discussed by a number of recent works (see for instance [36,37]).
Here we observe that the ramified instanton partition function is given by the same expressions (4.2) and (4.3) in which the function g(x) is [23,24,38]
g(x) = 2 sinhβx
2 (4.17)
Another difference with respect to the 2d/4d case is that the counting parameters qI are now dimensionless and are given by
qI = (−1)kI−1 βΛI bI for I = 1, . . . , M − 1 , qM = (−1)N βΛ 2N M −1 Y I=1 qI !−1 . (4.18)
The final result is obtained by summing the residues of z{dI} over the same set of poles
selected by the JK prescription (4.14).
Let us illustrate these ideas by calculating the twisted effective superpotential that governs the infrared behavior of the [1, 1] operator in SU(2). Up to two instantons, the partition function using these rules is
Zinst.[1, 1] = 1 +
q1
4 sinh β21 sinh β2(2a+1+ˆ2) +
q2
4 sinh β21 sinh β2(−2a+1+ˆ2)
+ q
2 1
16 sinh β21 sinh β1 sinh β2(2a + 1+ ˆ2) sinh β2(2a + 21+ ˆ2)
+ q
2 2
16 sinh β21 sinh β1 sinh β2(−2a + 1+ ˆ2) sinh β2(−2a + 21+ ˆ2)
+ q1q2 sinh
β
2(1+ 2ˆ2)
16 sinh2 β21 sinh β ˆ2 sinh β2(−2a + 1+ ˆ2) sinh β2(2a + ˆ2)
+ q1q2
16 sinh β21 sinh βˆ2 sinh β2(2a + 1+ ˆ2) sinh β2(2a + ˆ2)
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where a1 = −a2 = a. From this instanton partition function we can extract the twistedchiral superpotential in the usual manner according to (4.7). The result is
q1 dWinst dq1 = β 2 sinh(βa) Λ21+Λ 4 Λ21 −β 3cosh(βa) 8 sinh3(βa) Λ41+Λ 8 Λ41 + . . . (4.20)
It is very easy to check that in the limit β → 0 this expression reduces to the 2d/4d result in (4.12). Most importantly it agrees with the non-perturbative part of the solution (3.32) of the chiral ring equation of the 3d/5d SU(2) theory, thus confirming the validity of (3.27). Similar calculations can be performed for the higher rank cases without much difficulty, and indeed we have done these calculations for all surface operators of SU(4) and for many cases up to SU(6). Here, for brevity, we simply report the results at the one-instanton level for the surface operators in the SU(3) theory. In the case of the defect of type [1,2] the instanton partition function is
Zinst[1, 2] = 1 +
q1
8 sinh β21 sinh β2(a12+ 1+ ˆ2) sinh β2(a13+ 1+ ˆ2)
(4.21)
+ q2
8 sinh β21 sinh β2(−a12+ 1+ ˆ2) sinh β2(−a13+ 1+ ˆ2)
+ . . . ,
while for the defect of type [1,1,1] we find Zinst[1, 1, 1] = 1 + q1 4 sinh β21 sinh β2(a12+1+ˆ2) + q2 4 sinh β21 sinh β2(a23+ 1+ ˆ2) + q3 4 sinh β21 sinh β2(a31+1+ˆ2) + . . . (4.22)
where aij = a1 − aj. These expressions are clear generalizations of the 2d/4d instanton partition functions (4.15) and (4.16). Moreover one can check that the twisted superpo-tentials that can be derived from them perfectly match the ones obtained by solving the chiral ring equations as we discussed in section 3.
5 Superpotentials for dual quivers
The 2d/4d quiver theories considered in section 2 admit dual descriptions [7, 12, 13]. In particular, with repeated applications of Seiberg-like dualities, one can prove that the linear quiver of figure 1 is dual to the one represented in figure 5. Here the ranks of the U(rI) gauge groups are given by
rI = N − kI = M X
K=I+1
nK, (5.1)
where in the second step we have used (2.2) to express kI in terms of the entries of partition [n1, . . . , nM] labeling the surface defect. Notice the reversal of the arrows with respect to the quiver in figure1, and thus the different assignment of massive chiral fields to fundamental or anti-fundamental representations.
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N
r
1. . .
r
M −2r
M −1Figure 5. The quiver which is dual to the one in Fig-1.
The new quiver provides an alternative realization of the same surface operator [7]. Its corresponding twisted superpotential, which we denote by fW , is given by the obvious modification of (2.22), and reads11
f W = 2πi M −1 X I=1 rI X s=1 e τI(µ)eσ (I) s − M −1 X I=2 rI X s=1 rI−1 X t=1 $ eσ (I−1) t −σe (I) s − r1 X s=1 Tr $ Φ −eσ (1) s ) . (5.2)
As in (2.22), the linear terms ineτI(µ) are the classical contributions, while the other terms are the one-loop part. The dual FI couplings τeI(µ) renormalize like the orginal couplings τI(µ) but with kI replaced by rI. In view of (5.1), this implies that the one-loop β-function coefficient in the dual theory is opposite to that of the original theory, namely
ebI = rI+1− rI−1= −kI+1+ kI−1= −bI . (5.3) In turn, this implies that the dynamically generated scale in the I-th node of the dual theory is given by
e ΛbI
I = e
−2πiτeIµbI, (5.4)
to be compared with (2.27). As usual we can trade the couplings eτI(µ) for these scales e
ΛI, and thus rewrite the twisted superpotential (5.2) in a form that is the straightforward modification of (2.29).
If we make the following classical ansatz
e
σ(I)= diag(an1+...+nI+1, an1+...+nI+2, . . . , aN) , (5.5)
which is dual to the one for σ(I) given in (2.41), then it is easy to check that
Trσe(I) = −Tr σ(I) . (5.6)
This clearly implies
1 2πi ∂ fWclass ∂τeI = − 1 2πi ∂Wclass ∂τI . (5.7)
Thus, if the FI parameters in the two dual models are related to each other by
e
τI = −τI, (5.8)
11For later convenience, we denote the twisted chiral scalars and the FI couplings of the dual gauge groups
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3
2
Figure 6. The dual quiver for the SU(3)[1, 2] defect.
one has fWclass= Wclass. Notice that using (5.8) in (5.4) and comparing with (2.27), we have e
ΛI = ΛI . (5.9)
The relation (5.6) remains true also at the quantum level. This statement can be verified by expanding σe(I) as a power series in the various eΛI’s around the classical vac-uum (5.5), and iteratively solving the corresponding chiral ring equations in a semi-classical approximation. Doing this and using (5.8) and (5.9), we have checked the validity of (5.6) in several examples. Furthermore, we have obtained the same relations also using the localization methods described in section 4. Therefore, we can conclude that the two quiver theories in figure1and5, indeed provide equivalent descriptions of the 2d/4d defect SU(N )[n1, . . . , nM].
This conclusion changes drastically once we consider the 3d/5d quiver theories com-pactified on a circle. In this case, the dual superpotential corresponding to the quiver in figure 5, is obtained by upgrading (5.2) to a form analogous to (3.8), namely
f W = M −1 X I=1 rI X s=1 ebI log(β eΛI)σe(I)s − M −1 X I=2 rI X s=1 rI−1 X t=1 ` σe (I−1) t −eσ (I) s − r1 X s=1 Tr ` Φ −eσ (1) s . (5.10) Here we have used the loop-function `(x) defined in (3.3), and taken into account the renormalization of the FI couplings to introduce the scales eΛI. Using for the original quiver the ansatz (3.25), and for the dual theory the ansatz (5.5), which can be rewritten as
e
S(I)= diag(An1+...+nI+1, An1+...+nI+2, . . . , AN) (5.11)
in terms of the exponential variables eS(I)= eβσe (I)
and Ai= eβai, one can easily check that the relation (5.6) still holds true.
However, in general, this is no longer valid for the full solutions of the chiral ring equations. This happens whenever the ranks kI of the original quiver theory and the ranks rI of the dual model are different from each other for some I, which is the generic situation. Let us show this in a specific example, namely the defect of type [1,2] in the SU(3) theory. The original quiver theory was discussed in detail in section 3 where we have shown that the solution of the chiral ring equation is (see (3.36))
σ? = a1+ β2 A1/21 A12A13 Λ31+Λ 6 Λ31 + . . . . (5.12)
